Directed strongly regular graphs with rank 5

From the parameters (n, k, t, lambda, mu) of a directed strongly regular graph (dsrg) A. Duval (1988) [4] showed how to compute the eigenvalues and multiplicities of the adjacency matrix, and thus the rank of the adjacency matrix. For every rational number q, where 1/5 <= q <= 7/10 there is a feasible (i.e., satisfying Duval's conditions) parameter set for a dsrg with rank 5 and with k/n = q. In this paper we show that there exist a dsrg with such a feasible parameter set only if k/n is 1/5, 1/3, 2/5, 1/2, 3/5, or 2/3. Every dsrg with rank 5 therefore has parameters of a known graph. The proof is based on an enumeration of 5 x 5 matrices with entries in {0,1}. (C) 2015 Elsevier Inc. All rights reserved.